Weighted Sequences in Finite Cyclic Groups

Abstract

Let $p>7$ be a prime, let $G=\mathbb{Z}/p\mathbb{Z}$, and let $S_1={\prod }_{i=1}^p{g}_i$ and $S_2={\prod }_{i=1}^p{h}_i$ be two sequences with terms from $G$. Suppose that the maximum multiplicity of a term from either $S_1$ or $S_2$ is at most $\frac{2p+1}{5}$. Then we show that, for each $g\in G$, there exists a permutation $\sigma$ of $1,2,...,p$ such that $g={\sum }_{i=1}^p(g_i\cdot h_{\sigma (i)})$. The question is related to a conjecture of A. Bialostocki concerning weighted subsequence sums and the Erdős-Ginzburg-Ziv Theorem.